AVD, Android Video Downloader, is the best Android app to download your favorite videos.
Watch your videos whenever you want. With AVD, you can download videos when you have a good wi-fi connection and then enjoy your favorite video at your convenience. Even if you have no internet connection, you can check your videos on your device.
In the browser simply click on the video link that you wish to download. From the Action menu, select Video Downloader, and the download should start automatically. You should be able to see the download progress in the notification bar (at the top of the screen).
YouTube is not supported due to recent Google's lawsuits. Downloading materials from YouTube is clearly against its terms of service.
Downloads are automatically added to your Gallery and are stored into your "Download" directory (in this folder: /mnt/sdcard/Download). For Android 3.2 and above, downloads appears in your phone's download manager too.